I want to simplify this expression $im[{\hat {H}},{\hat {x}}]=\hbar {\hat {p}},\qquad i[{\hat {H}},{\hat {p}}]=-\hbar V'({\hat {x}}).$ to this $m{\frac {\partial H(x,p)}{\partial p}}=p,\qquad {\frac {\partial H(x,p)}{\partial x}}=V'(x),$.
I used conversion from position to momentum space, $[{\hat {H}},{\hat {x}}] = {\hat {H}}{\hat {x}} - {\hat {x}},{\hat {H}} = {\hat {H}}{i \hbar \frac{\partial}{\partial p}} - i \hbar \frac{\partial}{\partial p}}{\hat {H}}$ then I need to set $ \hat {H}}{i \hbar \frac{\partial}{\partial p}} = 0$. But it seems not formal!